Mit der Küchenrolle wische ich das Schneidebrett ab, bevor ich es neben das Spülbecken stelle.

Because mit always takes the dative case.

• Base form: die Küchenrolle
• After mit: mit der Küchenrolle

Küchenrolle is a feminine noun, so its dative singular article is der.

Die Küchenrolle usually means kitchen roll or paper towel roll.

In a sentence like this, it does not necessarily mean you are literally wiping with the whole roll. It often just means you are using paper towel from the roll.

Depending on context, German could also use:

• das Papiertuch = a paper towel / paper towel sheet
• das Küchentuch = a kitchen towel / dish towel

So mit der Küchenrolle is natural everyday German for using paper towel.

Because abwischen is a separable verb.

Its basic form is abwischen, but in a main clause the prefix separates:

• Ich wische das Schneidebrett ab.

Here:

• wische = the conjugated verb
• ab = the separable prefix

The prefix ab adds the idea of wiping off or wiping away.

In forms where the verb is not split, it stays together:

• infinitive: abwischen
• subordinate clause: ..., weil ich das Schneidebrett abwische

This is because German main clauses follow the V2 rule: the finite verb must be in second position.

The first position is taken by Mit der Küchenrolle, so the verb must come next:

• Mit der Küchenrolle | wische | ich | ...

That gives:

• Mit der Küchenrolle wische ich das Schneidebrett ab

You could also say:

• Ich wische das Schneidebrett mit der Küchenrolle ab

Both are correct. The version in your sentence gives a little more emphasis to how the action is done.

Both are compound nouns, and in German the last part usually determines the gender.

• das Brett → das Schneidebrett
• das Becken → das Spülbecken

So:

• Schneidebrett is neuter because Brett is neuter
• Spülbecken is neuter because Becken is neuter

This is a very useful rule for German compounds.

It is the direct object of abwischen, so it is in the accusative case.

With a neuter singular noun, the nominative and accusative article are the same:

• nominative: das Schneidebrett
• accusative: das Schneidebrett

So you do not see a form change here, even though the case is accusative.

Because bevor is a subordinating conjunction.

Subordinating conjunctions send the conjugated verb to the end of the clause:

• bevor ich es neben das Spülbecken stelle

This is normal German subordinate-clause word order.

Compare:

• Main clause: Ich stelle es neben das Spülbecken.
• Subordinate clause: ..., bevor ich es neben das Spülbecken stelle.

Es refers to das Schneidebrett.

Since Schneidebrett is neuter, the matching pronoun is es.

German, like English, often uses a pronoun instead of repeating the noun:

• ..., bevor ich das Schneidebrett neben das Spülbecken stelle
• ..., bevor ich es neben das Spülbecken stelle

The second version sounds more natural because it avoids repetition.

Because neben is a two-way preposition. It can take either:

• dative for location
• accusative for direction / destination

Here, the verb stellen expresses movement to a place, so German uses the accusative:

• ich stelle es neben das Spülbecken = I place it next to the sink

If you were describing where it already is, you would use the dative:

• Es steht neben dem Spülbecken = It is standing next to the sink

So the contrast is:

• motion toward a place → neben das Spülbecken
• existing location → neben dem Spülbecken

Possibly, depending on the exact situation.

German often distinguishes between different kinds of put:

• stellen = to set something upright / standing, or to place it somewhere
• legen = to lay something down flat
• setzen = to set something into a sitting position

With a cutting board:

• stellen suggests placing it so that it is upright or leaning
• legen suggests placing it flat

So if the board is being propped up next to the sink, stelle is a very good choice. If it were being laid flat on the counter, lege would sound more natural.